Context: The ionization fraction (i.e., the electron abundance) plays a key role in the chemistry and dynamics of molecular clouds. Aims: We study the H13CO^+, DCO+ and HOC+ line emission towards the Horsehead, from the shielded core to the UV irradiated cloud edge, i.e., the photodissociation region (PDR), as a template to investigate the ionization fraction gradient in molecular clouds. Methods: We analyze an IRAM Plateau de Bure Interferometer map of the H13CO+ J=1-0 line at a 6.8''× 4.7'' resolution, complemented with IRAM-30 m H13CO+ and DCO+ higher-J line maps and new HOC+ and CO+ observations. We compare self-consistently the observed spatial distribution and line intensities with detailed depth-dependent predictions of a PDR model coupled with a nonlocal radiative transfer calculation. The chemical network includes deuterated species, 13C fractionation reactions and HCO^+/HOC+ isomerization reactions. The role of neutral and charged PAHs in the cloud chemistry and ionization balance is investigated. Results: The detection of the HOC+ reactive ion towards the Horsehead PDR proves the high ionization fraction of the outer UV irradiated regions, where we derive a low [ HCO^+] /[ HOC^+] ≃ 75-200 abundance ratio. In the absence of PAHs, we reproduce the observations with gas-phase metal abundances, [Fe+Mg+...], lower than 4 × 10 -9 (with respect to H), and a cosmic-ray ionization rate of ζ=(5± 3)× 10 -17 s-1. The inclusion of PAHs modifies the ionization fraction gradient and increases the required metal abundance. Conclusions: The ionization fraction in the Horsehead edge follows a steep gradient, with a scale length of ~0.05 pc (or ~25''), from [ e^-] ≃ 10-4 (or n_e~˜ 1-5 cm-3) in the PDR to a few times ~10-9 in the core. PAH- anions play a role in the charge balance of the cold and neutral gas if substantial amounts of free PAHs are present ([PAH] > 10-8). Based on observations obtained with the IRAM Plateau de Bure interferometer and 30 m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).